Semi Substitivity

Terminology aside: The original term, as coined by Zeman, is
"Semisubstitutivity". I can neither reliably pronounce, nor
reliably type this. So hopefully people won't object too
strongly if I use the (easier for me) term "Semi-substitivity"

I'm sure that most readers of this are familiar with the concept in many
logics of "substitution of equivalences". It says (roughly) that if you have
p implying q, and q implying p, then you may substitute one for the other
anywhere.

It is natural, for people with the right world view, to ask the question:
"If p implying q, and q implying p, license replacing one for the other
anywhere in an expression, is there something that JUST p implying q
licenses you to do?" The short answer is yes, and the long answer is
that deep into an expression there are places where you may replace
p by q if you find a p. And even more surprising, all the other positions
in the expression where you have no license to replace p by q, you DO have
license to replace q by p if q is found.

While all positions in an expression tree that are in the scope of only
implicational operators obeying semi-substitivity license the substitution
one way or the other, which way is allowed at each position is somewhat
awkward to state, and is done below with the proof.

If we have a substitution instance of p > q (Call it p' > q') then
you can replace p' by q' anywhere you can replace p by q. And q' can be
replaced by p' anywhere q can be replaced by p.
This allows some local substitutions to be done without having to do
the substitution uniformly into the entire expression. (Substitute
into (p>q) and then proceed with semi-substitivity.)

To the best of my knowledge, the term "Semisubstitutivity" was coined by the
logician J. Jay Zeman in his paper "A system of Implicit Quantification" in
the Journal of Symbolic Logic, Volume 32 (1967) pages 480-504, where it was
introduced and played a somewhat minor role.

The concept achieved star billing the next year in his paper
"The Semisubstitutivity of Strict Implication", J. Jay. Zeman, Journal of
Symbolic Logic, Volume 33 # 3 (1968).
He mentions (but doesn't prove) that it is true in standard PC,
intuitionistic PC, and other logics, and explicitly constructs it for a
series of modal logics. The construction leans on the details of implication
in the systems in question, and parts of the discussion depended on
"counting the negations" when the implication was reduced to the modal
operators, negation, and "or". In my opinion this contributed to people
thinking of it in those terms, and not doing much with it in any more general
context.

These papers did not attract much attention. For example, to the best of my
my knowledge, the second paper didn't have a single paper reference it in the
40+ years since it came out.

Unknown to me when I started this page, the idea was independently
discovered by Anderson and Belnap and appears in volume 1 of their book
"Entailment" published in 1975. Judging by the Bibliography,
they were unaware of Zeman's work on the topc.
They refer to it as the Replacement Theorem.

Also unknown to me when I started this page, related ideas were independently
developed by Zohar Manna and Richard Waldinger who pursued the ideas
in an automated theorem proving context.
"Special Relations in Automated Deduction" [Journal of the ACM
(JACM), Volume 33, Issue 1, (January 1986) pages 1-59]. Note that they
use the term "substitutivity". I assume that they were independent
because their bibliography lists neither the Zeman works, nor Anderson
and Belnap's book.
They call their version Relation Replacement.

Proofs of substitution of equivalents often proceed from showing that if a
location in an expression is within the scope of only implications then the
substitutions can be done, to showing it is still true as you introduce
other operators.

This page will prove at locations in an expression that are in the scope of
any implicational operators in the system that have certain common properties.
Other operators can generally (but tediously) be shown to extend the properties
to their scope directly from the axioms that define them and the proof for
the implicational operator(s) of the system.

If you have uniform substitution, and you have one or more implicational
operators that obey the rule of Modus Ponens, and you have for each of
them the theorems or axioms of:

B: (p>q) > ((r>p) > (r>q)) {infix},
CCpqCCrpCrq {polish}

B': (p>q) > ((q>r) > (p>r)) {infix},
CCpqCCqrCpr {polish}

Then, by the proof given below, you have semi-substitivity for any location
in an expression that is in the scope of JUST those implicational operators.

Comments

Below we prove the theorems for the rules of semi-substitivity, and
because we have Modus Ponens, we therefore have the rules as well as
the theorems. The system MUST have modus ponens. There are "Equivalent"
systems with exactly the same theorems, but with rules weaker than modus ponens
(such as Jean Porte's version of PC in the 1960's and Lloyd Humberstone's systems
recently) that don't give us the rules from the theorems since they are too
weak to prove the rule of modus ponens.

Given p implies q, stating where p can be replaced by q (let's call this
forward replacement) and where q can be replaced by p (let's call this
reverse replacement) is somewhat awkward. As we look at the expression
tree, the top level node (by modus ponens) allows forward replacement.
As we go down the consequent (right) branch of a node the node below
inherits the direction of the parent. As we go down the antecedent (left)
branch the node has the opposite direction to its parent.

For polish notation in expressions with just the implicational operators
there is a very simple labeling of forward and reverse. The first operator
has forward replacement, the next token has reverse replacement, and it
alternates token wise down the string. [Token wise since multicharacter
variable names (like v12) need to be treated as a single item]. (To the best
of my knowledge, while I rediscovered it, the first place of publication appears
to be Anderson and Belnap's "Entailment" where it is attributed to
John Bacon.

As an example, let's take the expression (p > q) > (r > s)
and mark where each flavor of replacement can be done. (Using + for forward
replacement and - for reverse replacement.

Infix:
(p > q) > (r > s)
+ - - + - + +
Polish:
CCpqCrs
+-+-+-+

Modus Ponens is semi-substitivity at top level, B and B' are
semi-substitivity at the next level down, and theorems 3 had 5 extend
semi-substitivity down one level from whatever level we have already proven.
So by induction we can build the semi-substitivity rules to cover any finite
expression in the implicational operators meeting the assumptions.

Actually, only the first eight lines of the proof are strictly needed, but
I generated the next level for those people that are more comfortable with
more examples.